From c75ed79814a21224fabe17262227ad6a76f0917d Mon Sep 17 00:00:00 2001 From: Martin Thoma Date: Sun, 16 Feb 2014 11:46:14 +0100 Subject: [PATCH] Textsetzung; TODO entfernt; Kleine Korrekturen --- documents/GeoTopo/Abkuerzungen.tex | 1 + documents/GeoTopo/GeoTopo.pdf | Bin 5196753 -> 5197961 bytes documents/GeoTopo/Kapitel1-UB.tex | 5 +++ documents/GeoTopo/Kapitel1.tex | 4 +-- documents/GeoTopo/Kapitel3.tex | 5 +-- documents/GeoTopo/Kapitel4.tex | 30 ++++++++---------- documents/GeoTopo/Loesungen.tex | 20 ++++++++++-- .../GeoTopo/definitions/definitionen.pdf | Bin 359488 -> 359488 bytes .../GeoTopo/figures/coordinate-system-1.tex | 6 +++- .../GeoTopo/figures/coordinate-system-2.tex | 10 ++++-- documents/GeoTopo/meta/Arbeitszeit.md | 4 ++- .../GeoTopo/other-formats/GeoTopo-A5.pdf | Bin 5248270 -> 5249415 bytes .../coordinate-system-1.tex | 4 +++ .../coordinate-system-2.tex | 8 ++++- 14 files changed, 69 insertions(+), 28 deletions(-) diff --git a/documents/GeoTopo/Abkuerzungen.tex b/documents/GeoTopo/Abkuerzungen.tex index b6c366d..4b059e4 100644 --- a/documents/GeoTopo/Abkuerzungen.tex +++ b/documents/GeoTopo/Abkuerzungen.tex @@ -9,6 +9,7 @@ \acro{d. h.}{das heißt} \acro{Def.}{Definition} \acro{etc.}{et cetera} + \acro{Hom.}{Homomorphismus} \acro{o. 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zpFdhXd)n98M}Aye_EdR6{Jx{=XQqGEKIqp_XDB9rDf-#!6Hj)PJ?ve5!_slzOPv|4 zGEl{+o{b z40LX=y>(!1s*$icA#TX}^Go@J^EK>0h2|<8WaqT=_~6*YFH|$(K{JcIO3n{VF^a}06Ebmt^fc4 diff --git a/documents/GeoTopo/Kapitel1-UB.tex b/documents/GeoTopo/Kapitel1-UB.tex index b57d44b..4a02c11 100644 --- a/documents/GeoTopo/Kapitel1-UB.tex +++ b/documents/GeoTopo/Kapitel1-UB.tex @@ -54,3 +54,8 @@ \item ein Homöomorphismus, der kein Homomorphismus ist \end{bspenum} \end{aufgabe} + +\begin{aufgabe}[Begriffe]\label{ub3:meinsExtra2} + Definieren sie die Begriffe \enquote{Isomorphismus}, + \enquote{Isotopie} und \enquote{Isometrie}. +\end{aufgabe} diff --git a/documents/GeoTopo/Kapitel1.tex b/documents/GeoTopo/Kapitel1.tex index 165d8a0..b0e6a7d 100644 --- a/documents/GeoTopo/Kapitel1.tex +++ b/documents/GeoTopo/Kapitel1.tex @@ -230,7 +230,7 @@ Die Teilraumtopologie wird auch \textit{Spurtopologie} oder $\fB = \Set{\fB_r(x) \subseteq \powerset{X} | x \in X, r \in \mdr^+}$ ist Basis einer Topologie auf $X$. \end{bemerkung} -\begin{definition}\xindex{Isometrie}% +\begin{definition}\xindex{Isometrie}\label{def:Isometrie}% Seien $(X, d_X)$ und $(Y, d_Y)$ metrische Räume und $\varphi: X \rightarrow Y$ eine Abbildung mit \[\forall x_1, x_2 \in X: d_X(x_1, x_2) = d_Y(\varphi(x_1), \varphi(x_2)) \] @@ -1008,7 +1008,7 @@ $\qed$ \end{figure} \end{beispiel} -\begin{definition}\xindex{Knoten!äquivalente}\xindex{Isotopie}% +\begin{definition}\xindex{Knoten!äquivalente}\xindex{Isotopie}\label{def:Isotopie}% Zwei Knoten $\gamma_1, \gamma_2: S^1 \rightarrow \mdr^3$ heißen \textbf{äquivalent}, wenn es eine stetige Abbildung \[H: S^1 \times [0,1] \rightarrow \mdr^3\] diff --git a/documents/GeoTopo/Kapitel3.tex b/documents/GeoTopo/Kapitel3.tex index 772b78b..208a151 100644 --- a/documents/GeoTopo/Kapitel3.tex +++ b/documents/GeoTopo/Kapitel3.tex @@ -1194,12 +1194,13 @@ und der Fundamentalgruppe herstellen: g_1 \circ (g_2 \circ x) &= \varrho (g_1) (g_2 \circ x)\\ &= \varrho(g_1) (\varrho(g_2)(x))\\ &= (\varrho(g_1) \circ \varrho(g_2))(x)\\ - &\overset{\varrho \text {ist Hom.}}{=} \varrho(g_1 \cdot g_2) (x)\\ + &\overset{\mathclap{\varrho \text { ist Hom.}}}{=}\hspace{3 mm} \varrho(g_1 \cdot g_2) (x)\\ &= (g_1 \cdot g_2) \circ x \end{align*} z.~z. \cref{def:gruppenoperation.1}: - $1_G \cdot x = \varrho(1_G)(x) = \id_X(x) = x$, weil $\varrho$ Homomorphismus ist. + $1_G \cdot x = \varrho(1_G)(x) = \id_X(x) = x$, weil $\varrho$ ein + Homomorphismus ist. \end{beweis} \begin{beispiel}\label{bsp:13.4}%In Vorlesung: Beispiel 13.4 diff --git a/documents/GeoTopo/Kapitel4.tex b/documents/GeoTopo/Kapitel4.tex index eb030b6..b1e655d 100644 --- a/documents/GeoTopo/Kapitel4.tex +++ b/documents/GeoTopo/Kapitel4.tex @@ -429,7 +429,7 @@ schneiden sich. \label{fig:geometry-7} }% - \label{fig:formen} + \label{fig:winkel-und-parallelogramm} \caption{Situation aus \cref{bem:14.9}} \end{figure} @@ -678,15 +678,18 @@ $\xRightarrow{\text{Strahlensatz}} \frac{a}{h_c} = \frac{c}{h_a} \rightarrow a \ \item $(\mdr^2, d_\text{Euklid})$ ist offensichtlich eine euklidische Ebene. \item Sei $(X,d)$ eine euklidische Ebene und $g_1, g_2$ Geraden in $X$, die sich in einem Punkt $0$ im rechten Winkel - schneiden. Sei $X$ der Fußpunkt des Lots von $P$ auf - $g_1$ (vgl. \cref{ub11:aufg3.c}). + schneiden. - Sei $Y$ der Fußpunkt des Lots von $P$ auf $g_2$. + Sei $P \in X$ ein Punkt und $P_X$ der Fußpunkt des Lots von $P$ auf + $g_1$ (vgl. \cref{ub11:aufg3.c}) und $P_Y$ der Fußpunkt des Lots + von $P$ auf $g_2$. Setze $h(P) := (x_P, y_P)$ mit - $x_P := d(X, 0)$ und $y_P := d(Y, 0)$. + $x_P := d(P_X, 0)$ und $y_P := d(P_Y, 0)$. - \begin{figure}[ht] + In \cref{fig:14.13.0.1} wurde die Situation skizziert. + + \begin{figure}[htp] \centering \subfloat[Schritt 1]{ \resizebox{0.45\linewidth}{!}{\input{figures/coordinate-system-1.tex}} @@ -696,8 +699,8 @@ $\xRightarrow{\text{Strahlensatz}} \frac{a}{h_c} = \frac{c}{h_a} \rightarrow a \ \resizebox{0.45\linewidth}{!}{\input{figures/coordinate-system-2.tex}} \label{fig:14.13.2} }% - \label{fig:14.13.0.1} \caption{Beweis zu \cref{satz:14.13}} + \label{fig:14.13.0.1} \end{figure} Dadurch wird $h:X \rightarrow \mdr^2$ auf dem Quadranten @@ -716,18 +719,11 @@ $\xRightarrow{\text{Strahlensatz}} \frac{a}{h_c} = \frac{c}{h_a} \rightarrow a \ $P'$ auf der gleichen Seite von $g_2$ wie $P$. \end{beweis} \begin{beweis}[von 2] - \begin{figure}[ht] + \begin{figure}[htp] \centering - \subfloat[Schritt 1]{ - \resizebox{0.45\linewidth}{!}{\input{figures/coordinate-system-2.tex}} - \label{fig:14.13.3} - }% - \subfloat[Schritt 2 (Bild 13)]{ - \resizebox{0.45\linewidth}{!}{\input{figures/todo.tex}} - \label{fig:14.13.4} - }% - \label{fig:14.13.0.2} + \input{figures/coordinate-system-3.tex} \caption{Beweis zu \cref{satz:14.13}} + \label{fig:14.13.0.1} \end{figure} Zu Zeigen: $d(P, Q) = d(h(P), h(Q))$ diff --git a/documents/GeoTopo/Loesungen.tex b/documents/GeoTopo/Loesungen.tex index a51bf84..480a027 100644 --- a/documents/GeoTopo/Loesungen.tex +++ b/documents/GeoTopo/Loesungen.tex @@ -131,11 +131,11 @@ Die Definition von Homöomorphismus kann auf \cpageref{def:homoeomorphismus} nachgelesen werden. - \begin{definition} + \begin{definition}\xindex{Homomorphismus}% Seien $(G, *)$ und $(H, \circ)$ Gruppen und $\varphi:G \rightarrow H$ eine Abbildung. - $\varphi$ heißt \textbf{Homomorphismus}\xindex{Homomorphismus}, wenn + $\varphi$ heißt \textbf{Homomorphismus}, wenn \[\forall g_1, g_2 \in G: \varphi(g_1 * g_2) = \varphi(g_1) \circ \varphi(g_2)\] gilt. \end{definition} @@ -152,6 +152,22 @@ Kontexten verwendet. \end{solution} +\begin{solution}[\ref{ub3:meinsExtra2}] + Die Definition einer Isotopie kann auf \cpageref{def:Isotopie} nachgelesen + werden, die einer Isometrie auf \cpageref{def:Isometrie}. + + \begin{definition}\xindex{Isomorphismus}% + Seien $(G, *)$ und $(H, \circ)$ Gruppen und + $\varphi:G \rightarrow H$ eine Abbildung. + + $\varphi$ heißt \textbf{Isomorphismus}, wenn $\varphi$ ein bijektiver + Homomorphismus ist. + \end{definition} + + Eine Isotopie ist also für Knoten definiert, Isometrien machen nur in + metrischen Räumen Sinn und ein Isomorphismus benötigt eine Gruppenstruktur. +\end{solution} + \begin{solution}[\ref{ub4:aufg1}] \begin{enumerate}[label=(\alph*)] \item \textbf{Vor.:} Sei $M$ eine topologische Mannigfaltigkeit.\\ diff --git a/documents/GeoTopo/definitions/definitionen.pdf b/documents/GeoTopo/definitions/definitionen.pdf index be1e379271c7170292c57120f8a67f08ac03db38..90434788510c63c403eb96d86281b9928ce06819 100644 GIT binary patch delta 141 zcmX@`L-fE8(S{br7N!>F7M3lnu?3ojhQ`JQ2HFOO>IMetnq2z6`6(`mC8-J;E>=bc zh9(9^hGq~Y+uI6QOWB;w%+1Z6Ow27E4V+C~oy=TJObv|POifM9OiWD8-JDGA6l@47 L*=|?P%ESr)v`{08 delta 141 zcmX@`L-fE8(S{br7N!>F7M3lnu?3n2mL{gA#@Ysk>IMetnq2z6`6(`mC8-J;E>=bc zh9(9^hGq~Y+uI6QOWB-_Eu4&93=CWijNKd!EnEy;ErCjmOpP2(4Go=*ObqN4YzQgY KZdcCA#0mhlAR|Hm diff --git a/documents/GeoTopo/figures/coordinate-system-1.tex b/documents/GeoTopo/figures/coordinate-system-1.tex index 8ca868f..7703fda 100644 --- a/documents/GeoTopo/figures/coordinate-system-1.tex +++ b/documents/GeoTopo/figures/coordinate-system-1.tex @@ -2,6 +2,10 @@ \tkzSetUpPoint[shape=circle,size=10,color=black,fill=black] \tkzSetUpLine[line width=1] \tkzDefPoints{0/0/O, 1/0/X, 0/1/Y, 2/1/P} + + \tkzMarkAngle[fill=green!20,size=0.3cm,opacity=.5](X,O,Y) + \tkzLabelAngle[pos=0.15](X,O,Y){$\cdot$} + \tkzDrawLine[add=3 and 2](O,X) \tkzLabelLine[below,pos=3](O,X){$g_1$} \tkzLabelLine[right,pos=3](O,Y){$g_2$} @@ -10,4 +14,4 @@ \tkzLabelPoint(P){$P$} \node at ($(-2,2)$){$X$}; \tkzDrawPoints(P) -\end{tikzpicture} +\end{tikzpicture} \ No newline at end of file diff --git a/documents/GeoTopo/figures/coordinate-system-2.tex b/documents/GeoTopo/figures/coordinate-system-2.tex index 4e5212c..2d3aa52 100644 --- a/documents/GeoTopo/figures/coordinate-system-2.tex +++ b/documents/GeoTopo/figures/coordinate-system-2.tex @@ -2,6 +2,10 @@ \tkzSetUpPoint[shape=circle,size=10,color=black,fill=black] \tkzSetUpLine[line width=1] \tkzDefPoints{0/0/O, 1/0/X, 0/1/Y, 2/1/P} + + \tkzMarkAngle[fill=green!20,size=0.3cm,opacity=.5](X,O,Y) + \tkzLabelAngle[pos=0.15](X,O,Y){$\cdot$} + \tkzDrawLine[add=3 and 2](O,X) \tkzLabelLine[below,pos=3](O,X){$g_1$} \tkzLabelLine[right,pos=3](O,Y){$g_2$} @@ -23,6 +27,8 @@ \tkzLabelPoint[above right](P){$P$} \tkzLabelPoint[below left](O){$0$} + \tkzLabelPoint[below](xp){$P_X$} + \tkzLabelPoint[left](Y){$P_Y$} \node at ($(-2,2)$){$X$}; - \tkzDrawPoints(P) -\end{tikzpicture} + \tkzDrawPoints(P,Y,xp) +\end{tikzpicture} \ No newline at end of file diff --git a/documents/GeoTopo/meta/Arbeitszeit.md b/documents/GeoTopo/meta/Arbeitszeit.md index f82b3b2..7739228 100644 --- a/documents/GeoTopo/meta/Arbeitszeit.md +++ b/documents/GeoTopo/meta/Arbeitszeit.md @@ -82,4 +82,6 @@ in dem Erstellen dieses Skripts steckt: |14.02.2014 | 06:15 - 07:10 | Verbesserungsvorschläge von Arthur (Email) umgesetzt. |14.02.2014 | 18:30 - 18:50 | Verbesserungsvorschläge von Henrieke (RL) umgesetzt. |14.02.2014 | 18:50 - 19:00 | Verbesserungsvorschläge von Jan (Facebook) umgesetzt. -|14.02.2014 | 20:30 - 24:00 | Überarbeitung des Beweises der Eindeutigkeit einer Parallelen \ No newline at end of file +|14.02.2014 | 20:30 - 24:00 | Überarbeitung des Beweises der Eindeutigkeit einer Parallelen +|15.02.2014 | 13:00 - 21:00 | Textsetzung; Kleine Korrekturen +|16.02.2014 | 10:30 - 11:30 | 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zZrd?bvhwUE9$y_Ll8!tg&8v`~&$u;G4)Zwbf3NheH^F(Tz~b>BwtjzoltKDrZdYl{ zrJJ`^qm*M1i}YW$OkoEKtxB^8lMGKwEJQuxk16wR%3hfX3vv8B@WRz8unyH@we+u9vG8PME{+Qt(>4OkrSQdO7BS?@0*MeL!F6a`Z_nLC_6ri%}=8!<)E*;apB7=Da2{KPJ@;s^fdKRKuS+9 z-fua2;s^z(lQ3g_daOaj)*|%Z1#=9K9sj zWN0zOh;=sq%Lwpq(*7rsfPZ~y|Bd1+Nt~rGceeLh@A!C}`&UxNzqv)0XHhRbemeq) c?H#G9$WvrEu$!F+$;rz|Qwa!Yo9a;gFZ<1$;Q#;t diff --git a/tikz/coordinate-system-1/coordinate-system-1.tex b/tikz/coordinate-system-1/coordinate-system-1.tex index 2da01d7..f5bf0bd 100644 --- a/tikz/coordinate-system-1/coordinate-system-1.tex +++ b/tikz/coordinate-system-1/coordinate-system-1.tex @@ -9,6 +9,10 @@ \tkzSetUpPoint[shape=circle,size=10,color=black,fill=black] \tkzSetUpLine[line width=1] \tkzDefPoints{0/0/O, 1/0/X, 0/1/Y, 2/1/P} + + \tkzMarkAngle[fill=green!20,size=0.3cm,opacity=.5](X,O,Y) + \tkzLabelAngle[pos=0.15](X,O,Y){$\cdot$} + \tkzDrawLine[add=3 and 2](O,X) \tkzLabelLine[below,pos=3](O,X){$g_1$} \tkzLabelLine[right,pos=3](O,Y){$g_2$} diff --git a/tikz/coordinate-system-2/coordinate-system-2.tex b/tikz/coordinate-system-2/coordinate-system-2.tex index 8c66aba..a97f247 100644 --- a/tikz/coordinate-system-2/coordinate-system-2.tex +++ b/tikz/coordinate-system-2/coordinate-system-2.tex @@ -10,6 +10,10 @@ \tkzSetUpPoint[shape=circle,size=10,color=black,fill=black] \tkzSetUpLine[line width=1] \tkzDefPoints{0/0/O, 1/0/X, 0/1/Y, 2/1/P} + + \tkzMarkAngle[fill=green!20,size=0.3cm,opacity=.5](X,O,Y) + \tkzLabelAngle[pos=0.15](X,O,Y){$\cdot$} + \tkzDrawLine[add=3 and 2](O,X) \tkzLabelLine[below,pos=3](O,X){$g_1$} \tkzLabelLine[right,pos=3](O,Y){$g_2$} @@ -31,7 +35,9 @@ \tkzLabelPoint[above right](P){$P$} \tkzLabelPoint[below left](O){$0$} + \tkzLabelPoint[below](xp){$P_X$} + \tkzLabelPoint[left](Y){$P_Y$} \node at ($(-2,2)$){$X$}; - \tkzDrawPoints(P) + \tkzDrawPoints(P,Y,xp) \end{tikzpicture} \end{document}